Integration and nonlinear transformations in Hilbert space

Definition2. A function,/, from 77 to a topological vector space is uniformly continuous near zero in the topology 772 if there exists a sequence, An, of Hilbert-Schmidt operators such that ||.4B||2-»0 and such that / is uniformly continuous in the topology 772 on each of the sets U" = {x:\\\<l}.Remark.If 77 is finite dimensional then the preceding definition is equivalent to ordinary continuity.Indeed if / is continuous then it is uniformly continuous on the sphere of radius n about the origin, i.e., one may take An=n~1I where 7 is the identity operator on 77.